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Article

MSC: 06B05, 06D30
Keywords:
Antitone involution; De Morgan laws; Kleene algebra; distributive lattice; pseudo-Kleene algebra; subdirectly irreducible
Summary:
We introduce the concept of a pseudo-Kleene algebra which is a non-distributive modification of a Kleene algebra introduced by J. A. Kalman [Kalman, J. A.: Lattices with involution. Trans. Amer. Math. Soc. 87 (1958), 485–491.]. Basic properties of pseudo-Kleene algebras are studied. For pseudo-Kleene algebras with a fix-point there are determined subdirectly irreducible members.
References:
[1] Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloqnium Publ. 25, second edition, American Mathematical Society, New York, 1948. MR 0029876 | Zbl 0033.10103
[2] Cignoli, R.: Injective de Morgan and Kleene algebras. Proc. Amer. Math. Soc. 47 (1975), 269–278. DOI 10.1090/S0002-9939-1975-0357259-4 | MR 0357259 | Zbl 0301.06009
[3] Crvenkovič, S., Madarasz, R. Sz.: On Kleene algebras. Theoret. Comput. Sci. 108 (1993), 17–24. DOI 10.1016/0304-3975(93)90228-L | MR 1203820 | Zbl 0778.03006
[4] Kalman, J. A.: Lattices with involution. Trans. Amer. Math. Soc. 87 (1958), 485–491. DOI 10.1090/S0002-9947-1958-0095135-X | MR 0095135 | Zbl 0228.06003
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